After these preliminaries, the paper can be broken into three rough parts: constructing the family of functions \(f=f_{i,A,\rho }\)  and graphs \(\Gamma =\Gamma _{i,A,\rho }\)  that we will use in Theorems REF  and REF , proving lower bounds on the \(\beta \) –numbers of these surfaces, and estimating the Riesz transform on these surfaces. In Section , we construct a family of intrinsic Lipschitz graphs based on the construction in [1]}. These graphs have bumps at many different scales, and in Section , we calculate the effect of these bumps on the \(\beta \) –numbers and prove Theorem REF .
